WP_Query meta compare must include ALL array values

You can have multiple meta_query arguments with the same key and that will produce the results you’re seeking. Here’s an example:

$meta_query = array_map( create_function( '$a', 'return array("key" => "row_amenities", "value" => $a);' ), $amenities );

$the_query = new WP_Query(array(
            'post_type' => 'listing',
            'posts_per_page' => -1,
            'post_status' => 'publish', 
            'category_name' => 'private_rental',
            'orderby' => 'rand',
            'meta_query' => $meta_query
        ));

If $amenities will have a lot of values (> 10), you should probably take a different approach, because it will join the postmeta table for each meta_query.